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Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Optimal฀Möbius฀Transformation for฀Information฀Visualization฀and฀Meshing

Marshall฀Bern

Xerox฀Palo฀Alto฀Research฀Ctr.

David฀Eppstein

Univ.฀of฀California,฀Irvine Dept.฀of฀Information฀and฀Computer฀Science

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

What฀are฀Möbius฀transformations?

Fractional฀linear฀transformations฀of฀complex฀numbers:

z฀→฀(a฀z฀+฀b)฀/฀(c฀z฀+฀d)

But฀what฀does฀it฀mean฀geometrically? How฀to฀generalize฀to฀higher฀dimensions? What฀is฀it฀good฀for?

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

What฀are฀Möbius฀transformations?฀฀(II)

Circle-preserving฀maps฀from฀the฀plane฀to฀itself More฀intuitive Generalizes฀nicely฀to฀spheres,฀higher฀dimensional฀spaces Not฀very฀concrete

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

What฀are฀Möbius฀transformations฀฀(III)

Inversion:฀map฀radii฀of฀circle฀to฀same฀ray so฀that฀product฀of฀distances฀from฀center฀=฀radius2 Möbius฀transformation฀=฀composition฀of฀multiple฀inversions More฀concrete,฀still฀generalizes฀nicely

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

What฀are฀Möbius฀transformations?฀฀(IV)

View฀plane฀as฀boundary฀of฀halfspace฀model฀of฀hyperbolic฀space Möbius฀transformations฀of฀plane฀↔฀hyperbolic฀isometries Esoteric Most฀useful฀for฀our฀algorithms

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Optimal฀Möbius฀transformation:

Given฀a฀planar฀(or฀higher฀dimensional)฀input฀confjguration Select฀a฀Möbius฀transformation from฀the฀(six-dimensional฀or฀higher)฀space฀of฀all฀Möbius฀transformations That฀optimizes฀the฀shape฀of฀the฀transformed฀input Typically฀min-max฀or฀max-min฀problems: maximize฀min(set฀of฀functions฀describing฀transformed฀shape฀quality)

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Application:฀spherical฀graph฀drawing

Theorem฀[Koebe,฀Thurston]:฀Any฀planar฀graph฀can฀be฀represented by฀disjoint฀disks฀on฀a฀sphere฀so฀two฀vertices฀adjacent฀iff฀two฀disks฀tangent For฀maximal฀planar฀graphs,฀unique฀up฀to฀Möbius฀transformation Other฀graphs฀can฀be฀made฀maximal฀planar฀by฀adding฀vertex฀in฀each฀face

Optimization฀problem:

Find฀disk฀representation฀of฀G฀maximizing฀minimum฀disk฀radius

• r,฀given฀one฀disk฀representation,fjnd฀Möbius฀transformation฀maximizing฀min฀radius

Solution฀also฀turns฀out฀to฀display฀all฀symmetries฀of฀initial฀embedded฀graph

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Application:฀conformal฀mesh฀generation

Given฀simply-connected฀planar฀domain฀to฀be฀meshed Map฀to฀square,฀use฀regular฀mesh,฀invert฀map฀to฀give฀mesh฀in฀original฀domain Different฀points฀of฀domain฀may฀have฀different฀requirements฀for฀element฀size To฀minimize฀#฀elements,฀map฀regions฀requiring฀small฀size฀to฀large฀areas฀of฀square Conformal฀map฀is฀unique฀up฀to฀Möbius฀transformation

Optimization฀Problem:

Find฀conformal฀map฀maximizing฀min(size฀requirement฀*฀local฀expansion฀factor) to฀minimize฀overall฀number฀of฀elements฀produced

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Application:฀hyperbolic฀browser฀[Lamping,฀Rao,฀and฀Pirolli,฀1995]

Technique฀for฀viewing฀large฀web฀sites฀or฀other฀structured฀information by฀laying฀out฀information฀in฀hyperbolic฀space Allows฀“fjsheye฀view”:฀close-up฀look฀at฀details฀of฀some฀point฀in฀site global฀structure฀of฀site฀visible฀towards฀boundary฀of฀hyperbolic฀model The฀farther฀away฀a฀point฀is฀from฀the฀viewpoint฀(in฀hyperbolic฀distance) the฀smaller฀the฀information฀it฀represents฀will฀be฀displayed

Optimization฀problem:

Find฀good฀initial฀viewpoint฀for฀hyperbolic฀browser in฀order฀to฀make฀overall฀site฀as฀visible฀as฀possible Maximize฀minimum฀size฀of฀displayed฀object

• r

Maximize฀minimum฀separation฀between฀two฀objects

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Application:฀brain฀fmat฀mapping฀[Hurdal฀et฀al.฀1999]

Problem:฀visualize฀the฀human฀brain All฀the฀interesting฀stuff฀is฀on฀the฀surface But฀diffjcult฀because฀the฀surface฀has฀complicated฀folds Approach:฀fjnd฀quasi-conformal฀mapping฀brain฀→฀plane Then฀can฀visualize฀brain฀functional฀units฀as฀regions฀of฀mapped฀plane Avoids฀distorting฀angles฀but฀areas฀can฀be฀greatly฀distorted As฀in฀mesh฀gen.฀problem,฀mapping฀unique฀up฀to฀Möbius฀transformation

Optimization฀problem:

Given฀map฀3d฀triangulated฀surface฀→฀plane, fjnd฀Möbius฀transformation฀minimizing฀max(area฀distortion฀of฀triangle)

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Optimal฀Möbius฀Algorithm Key฀components:

Quasiconvex฀Programming Hyperbolic฀Geometry

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Quasiconvex฀functions

Level฀sets฀are฀nested฀convex฀curves Inner฀curves฀correspond฀to฀smaller฀function฀values (Like฀topographic฀map฀of฀open฀pit฀mine)฀

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Quasiconvex฀programming฀[Amenta,฀Bern,฀Eppstein฀1999]

Given฀n฀quasiconvex฀functions฀fi max(fi(x))฀is฀also฀quasiconvex problem฀is฀simply฀to฀compute฀x฀minimizing฀max(fi(x)) Can฀be฀solved฀exactly฀with฀O(n)฀constant-size฀subproblems using฀low-dimensional฀linear-programming-type฀techniques Can฀be฀solved฀numerically฀by฀hill-climbing฀or฀other฀local฀optimization฀methods

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Hyperbolic฀space฀(Poincaré฀model)

Interior฀of฀unit฀sphere;฀lines฀and฀planes฀are฀spherical฀patches฀perpendicular฀to฀unit฀sphere

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Two฀models฀of฀hyperbolic฀space Klein฀Model

Hyperbolic฀objects฀are฀straight฀or฀convex iff฀their฀model฀is฀straight฀or฀convex Angles฀are฀severely฀distorted Hyperbolic฀symmetries฀are฀modeled฀as Euclidean฀projective฀transformations

Poincaré฀Model

Angles฀in฀hyperbolic฀space equal฀Euclidean฀angles฀of฀their฀models Straightness/convexity฀distorted Hyperbolic฀symmetries฀are฀modeled฀as Möbius฀transformations

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

฀Möbius฀transformation฀and฀hyperbolic฀geometry

Quasiconvex฀programming฀works฀equally฀well฀in฀hyperbolic฀space Due฀to฀convexity-preserving฀properties฀of฀Klein฀model But฀space฀of฀Möbius฀transformations฀is฀not฀hyperbolic฀space...

View฀Möbius฀transformation฀as฀choice฀of฀Poincaré฀model

Factor฀transformations฀into choice฀of฀center฀point฀in฀hyperbolic฀model฀(affects฀shape) Euclidean฀rotation฀around฀center฀point฀(doesn’t฀affect฀shape)

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Optimal฀Möbius฀transformation฀algorithm

Represent฀optimization฀problem฀objective฀function as฀max฀of฀set฀of฀quasiconvex฀functions where฀function฀argument฀is฀hyperbolic฀center฀point฀location Hard฀part:฀proving฀that฀our฀objective฀functions฀are฀quasiconvex Solve฀quasiconvex฀program Use฀center฀point฀location฀to฀fjnd฀Möbius฀transformation

Optimal฀Möbius฀Transformation D.฀Eppstein,฀UC฀Irvine,฀WADS฀2001

Conclusions

Formulate฀several฀interesting฀applications฀as฀Möbius฀optimization Can฀solve฀via฀LP-type฀techniques฀or฀hill-climbing Interesting฀use฀of฀hyperbolic฀methods฀in฀computational฀geometry

but...

Details฀of฀exact฀algorithm฀may฀be฀diffjcult฀to฀implement (see฀Gärtner฀for฀similar฀diffjculties฀in฀LP-type฀min-volume฀ellipsoid) Not฀able฀to฀prove฀quasiconvexity฀in฀some฀cases e.g.฀given฀number฀x,฀triangle฀T฀at฀infjnity฀in฀hyperbolic฀3-space are฀points฀from฀which฀T฀subtends฀solid฀angle฀>฀x฀convex?